Problem: $P(x)$ is a polynomial. $P(x)$ divided by $(x-9)$ has a remainder of $1$. $P(x)$ divided by $(x-4)$ has a remainder of $7$. $P(x)$ divided by $(x+4)$ has a remainder of $0$. $P(x)$ divided by $(x+9)$ has a remainder of $-5$. Find the following values of $P(x)$. $P(4)=$
Explanation: We can use the polynomial remainder theorem to solve this problem: For a polynomial $p(x)$ and a number $a$, the remainder on division by $x-a$ is $p(a)$. According to the theorem, $P({4})$ is equal to the remainder when $P(x)$ is divided by $(x-{4})$, and we are given that this remainder is equal to $7$. In a similar manner, $P({-9})$ is equal to the remainder when $P(x)$ is divided by $(x-({-9}))$, which can be rewritten as $(x+9)$, and we are given that this remainder is equal to $-5$. In conclusion, $P(4)=7$ $P(-9)=-5$